As the finite discrete kernels are trace class and the \(\mathcal {A}^{\pm }_{2 \rightarrow 1}\)  kernels are conjugate to such, a computation similar to that of the proof of Theorem REF  (see, e.g., [1]}) shows that if \(a = 2 \theta + \theta ^{1/3} s\)  then
\((\mathrm {tr}K)_{\ell ^2 \lbrace a, a+1, \dots \rbrace } = \sum _{b \ge a} K_{sp/o}(b, b) \rightarrow \text{``} (\mathrm {tr}\mathcal {A}^{\pm }_{2 \rightarrow 1})_{L^2 (s, \infty )} = \int _s^\infty \mathcal {A}^{\pm }_{2 \rightarrow 1}(x,x) \mathrm {d}x \text{''}\) 
